The Inclusive Two Jet Triply Differential Cross Section

TL;DR

This study analyzes the inclusive two-jet triply differential cross section $d^3\sigma/dE_T d\eta_1 d\eta_2$ at Fermilab energies to probe parton densities and hard-scattering dynamics. It presents both the LO and full NLO QCD predictions, highlighting how the cross section’s shape, especially in central pseudorapidity, constrains gluon densities at small to moderate $x$ and how scale choices affect normalization and stability. By examining specific slices (D0 signed distribution and CDF SS/OS ratio), the work demonstrates the PDF sensitivity and the potential of experimental data to discriminate between PDF sets, while acknowledging normalization uncertainties. The central region emerges as particularly informative for PDF extraction, with NLO corrections improving reliability and enabling more precise comparisons with Tevatron data.

Abstract

We study the inclusive two jet triply differential cross section $d^3σ/dE_Tdη_1dη_2$ at Fermilab energies. Different $η_1$ and $η_2$ pseudorapidity regions are directly related to both the parton level matrix elements and the parton densities at leading order. We present the next-to-leading order [${\cal O}(α_s^3)$] corrections and show that the shape of the distribution at fixed transverse energy $E_T$ is a particularly powerful tool for constraining the parton distributions at small to moderate $x$ values. We investigate the renormalisation/factorisation scale uncertainty present in the normalisation and shape of the distribution at next-to-leading order. We discuss specific slices of the distribution, the same-side/opposite side ratio and the signed pseudorapidity distribution, in detail and compare them with preliminary experimental data.

Figures (18)

• Figure 1: The phase space boundary in the $\eta_1-\eta_2$ plane at leading-order (solid) and next-to-leading order (dotted) for $E_T = 50$ GeV and $\sqrt{s} = 1800$ GeV. The dashed line separates the 'small' $x$ -- 'large' $x$ and 'large' $x$ -- 'large' $x$ regions. In region I, either $x_1$ or $x_2$ is less than $x_T$, while in region II, both $x_1$ and $x_2$ are bigger than $x_T$. Region III is only permitted at next-to-leading order.
• Figure 2: Contours of (a) constant $\eta_1$ and (b) constant $\eta^*$ in the $x_1-x_2$ plane for $E_T = 50$ GeV and $\sqrt{s} = 1800$ GeV.
• Figure 3: (a) The ratio of the 'single effective parton density' of Eq. 4 for the MRSD${}_0$ and MRSD${}_{-}$ distributions compared to the MRSA parameterisation at $\mu= 50$ GeV. (b) The ratio of the gluon and quark parton densities in the MRSD${}_0$ distribution compared to the MRSA parameterisation at the same scale.
• Figure 4: The parton-parton luminosity for the MRSD${}_0$ and MRSD${}_{-}$ parton densities in the 'single effective subprocess approximation' as a function of $\eta_{\rm boost}$ for $|\eta^*| = 0, 1$ and 2 and $\mu = 50$ GeV.
• Figure 5: The lowest order squared matrix elements $|{\cal M}_{gg}|^2/\cosh^4(\eta^*)$ for $gg\to gg$ scattering as a function of $\eta^*$.